Topology, Geometry, and Mechanics of Z-Plasty

Reconstructive surgeries often use topological manipulation of tissue to minimize postoperative scarring. The most common version of this, Z-plasty, involves modifying a straight line cut into a Z shape, followed by a rotational transposition of the resulting triangular pedicle flaps, and a final restitching of the wound. This locally reorients the anisotropic stress field and reduces the potential for scarring. We analyze the planar geometry and mechanics of the Z-plasty to quantify the rotation of the overall stress field and the local forces on the restitched cut using theory, simulations, and simple physical Z-plasty experiments with foam sheets that corroborate each other. Our study rationalizes the most typical surgical choice of this angle, and opens the way for a range of surgical decisions by characterizing the stresses along the cut.

Organ size control via hydraulically gated oscillations

Hollow vesicular tissues of various sizes and shapes arise in biological organs such as ears, guts, hearts, brains and even entire organisms. Regulating their size and shape is crucial for their function. Although chemical signaling has been thought to play a role in the regulation of cellular processes that feed into larger scales, it is increasingly recognized that mechanical forces are involved in the modulation of size and shape at larger length scales. Motivated by a variety of examples of tissue cyst formation and size control that show simultaneous growth and size oscillations, we create a minimal theoretical framework for the growth and dynamics of a soft, fluid-permeable, spherical shell. We show that these shells can relieve internal pressure by bursting intermittently, shrinking and re-growing, providing a simple mechanism by which hydraulically gated oscillations can regulate size. To test our theory, we develop an in vitro experimental set-up to monitor the growth and oscillations of a hollow tissue spheroid growing freely or when confined. A simple generalization of our theory to account for irreversible deformations allows us to explain the time scales and the amplitudes of oscillations in terms of the geometry and mechanical properties of the tissue shells. Taken together, our theory and experimental observations show how soft hydraulics can regulate the size of growing tissue shells.

Morphogenesis one century after On Growth and Form

Summary: This Editorial introduces the special issue – providing a perspective on the influence of D'Arcy Thompson's work and an overview of the articles in this issue.

On the growth and form of shoots

Growing plant stems and shoots exhibit a variety of shapes that embody growth in response to various stimuli. Building on experimental observations, we provide a quantitative biophysical theory for these shapes by accounting for the inherent observed passive and active effects: (i) the active controllable growth response of the shoot in response to its orientation relative to gravity, (ii) proprioception, the shoot's growth response to its own observable current shape, and (iii) the passive elastic deflection of the shoot due to its own weight, which determines the current shape of the shoot. Our theory separates the sensed and actuated variables in a growing shoot and results in a morphospace diagram in terms of two dimensionless parameters representing a scaled local active gravitropic sensitivity, and a scaled passive elastic sag. Our computational results allow us to explain the variety of observed transient and steady morphologies with effective positive, negative and even oscillatory gravitropic behaviours, without the need for ad hoc complex spatio-temporal control strategies in terms of these parameters. More broadly, our theory is applicable to the growth of soft, floppy organs where sensing and actuation are dynamically coupled through growth processes via shape.

Grasping with a soft glove: intrinsic impedance control in pneumatic actuators

The interaction of a robotic manipulator with unknown soft objects represents a significant challenge for traditional robotic platforms because of the difficulty in controlling the grasping force between a soft object and a stiff manipulator. Soft robotic actuators inspired by elephant trunks, octopus limbs and muscular hydrostats are suggestive of ways to overcome this fundamental difficulty. In particular, the large intrinsic compliance of soft manipulators such as 'pneu-nets'-pneumatically actuated elastomeric structures-makes them ideal for applications that require interactions with an uncertain mechanical and geometrical environment. Using a simple theoretical model, we show how the geometric and material nonlinearities inherent in the passive mechanical response of such devices can be used to grasp soft objects using force control, and stiff objects using position control, without any need for active sensing or feedback control. Our study is suggestive of a general principle for designing actuators with autonomous intrinsic impedance control.

Excitable Dynamics and Yap-Dependent Mechanical Cues Drive the Segmentation Clock

The periodic segmentation of the vertebrate body axis into somites, and later vertebrae, relies on a genetic oscillator (the segmentation clock) driving the rhythmic activity of signaling pathways in the presomitic mesoderm (PSM). To understand whether oscillations are an intrinsic property of individual cells or represent a population-level phenomenon, we established culture conditions for stable oscillations at the cellular level. This system was used to demonstrate that oscillations are a collective property of PSM cells that can be actively triggered in vitro by a dynamical quorum sensing signal involving Yap and Notch signaling. Manipulation of Yap-dependent mechanical cues is sufficient to predictably switch isolated PSM cells from a quiescent to an oscillatory state in vitro, a behavior reminiscent of excitability in other systems. Together, our work argues that the segmentation clock behaves as an excitable system, introducing a broader paradigm to study such dynamics in vertebrate morphogenesis.

Controllable biomimetic birdsong

Birdsong is the product of the controlled generation of sound embodied in a neuromotor system. From a biophysical perspective, a natural question is that of the difficulty of producing birdsong. To address this, we built a biomimetic syrinx consisting of a stretched simple rubber tube through which air is blown, subject to localized mechanical squeezing with a linear actuator. A large static tension on the tube and small dynamic variations in the localized squeezing allow us to control transitions between three states: a quiescent state, a periodic state and a solitary wave state. The static load brings the system close to threshold for spontaneous oscillations, while small dynamic loads allow for rapid transitions between the states. We use this to mimic a variety of birdsongs via the slow-fast modulated nonlinear dynamics of the physical substrate, the syrinx, regulated by a simple controller. Finally, a minimal mathematical model of the system inspired by our observations allows us to address the problem of song mimicry in an excitable oscillator for tonal songs.

Controlling the Roughness of Langmuir-Blodgett Monolayers

Controlling the surface roughness of thin films with nanoscale precision is of significant interest for the rational design of surface coatings. Although wrinkling and buckling of Langmuir monolayers under compression has been demonstrated for several years, there is currently no method to precisely control this behavior during compression and thereby modify the surface roughness of deposited films. Here, we combine conventional Langmuir phase analysis with a novel dynamic viscoelasticity measurement to simply and accurately observe the jamming transition of monolayers of silica spheres, graphene oxide, and surfactant. By overcompressing beyond this point, the surface roughness of the deposited monolayer can be precisely controlled. This technique could be used to tune the surface properties of a variety of materials from lipids to nanoparticles.

Optimal strategies for throwing accurately

The accuracy of throwing in games and sports is governed by how errors in planning and initial conditions are propagated by the dynamics of the projectile. In the simplest setting, the projectile path is typically described by a deterministic parabolic trajectory which has the potential to amplify noisy launch conditions. By analysing how parabolic trajectories propagate errors, we show how to devise optimal strategies for a throwing task demanding accuracy. Our calculations explain observed speed-accuracy trade-offs, preferred throwing style of overarm versus underarm, and strategies for games such as dart throwing, despite having left out most biological complexities. As our criteria for optimal performance depend on the target location, shape and the level of uncertainty in planning, they also naturally suggest an iterative scheme to learn throwing strategies by trial and error.

Phototactic guidance of a tissue-engineered soft-robotic ray

Inspired by the relatively simple morphological blueprint provided by batoid fish such as stingrays and skates, we created a biohybrid system that enables an artificial animal--a tissue-engineered ray--to swim and phototactically follow a light cue. By patterning dissociated rat cardiomyocytes on an elastomeric body enclosing a microfabricated gold skeleton, we replicated fish morphology at 1/10 scale and captured basic fin deflection patterns of batoid fish. Optogenetics allows for phototactic guidance, steering, and turning maneuvers. Optical stimulation induced sequential muscle activation via serpentine-patterned muscle circuits, leading to coordinated undulatory swimming. The speed and direction of the ray was controlled by modulating light frequency and by independently eliciting right and left fins, allowing the biohybrid machine to maneuver through an obstacle course.

Spontaneous exfoliation of a drying gel

Wet starch cracks when it dries inhomogeneously, while hot glass cracks when it cools non-uniformly. In both cases, differential shrinkage induced by drying/cooling from the surface causes superficial cracks to grow perpendicular to the surface in different patterns. In contrast with these observations of bulk cracking in brittle materials, when a soft and homogeneously swollen polymer gel dries, differential strains lead to the peeling of a thin layer that spontaneously tears away from the bulk. Continued drying leads to the process repeating itself, forming a peeled-layered structure. The emergent thickness of the exfoliated layer is a function of both the geometry of the original gel and the physical parameters associated with the drying rate and external temperature. We characterize the experimental conditions under which layer peeling can arise, and use simulations to corroborate these observations. Finally, a minimal theory explains the scaling of the peel thickness, consistent with our experiments.

Integrative neuromechanics of crawling in D. melanogaster larvae

Locomotion in an organism is a consequence of the coupled interaction between brain, body and environment. Motivated by qualitative observations and quantitative perturbations of crawling in Drosophila melanogaster larvae, we construct a minimal integrative mathematical model for its locomotion. Our model couples the excitation-inhibition circuits in the nervous system to force production in the muscles and body movement in a frictional environment, thence linking neural dynamics to body mechanics via sensory feedback in a heterogeneous environment. Our results explain the basic observed phenomenology of crawling with and without proprioception, and elucidate the stabilizing role that proprioception plays in producing a robust crawling phenotype in the presence of biological perturbations. More generally, our approach allows us to make testable predictions on the effect of changing body-environment interactions on crawling, and serves as a step in the development of hierarchical models linking cellular processes to behavior.

Optimal switching between geocentric and egocentric strategies in navigation

Animals use a combination of egocentric navigation driven by the internal integration of environmental cues, interspersed with geocentric course correction and reorientation. These processes are accompanied by uncertainty in sensory acquisition of information, planning and execution. Inspired by observations of dung beetle navigational strategies that show switching between geocentric and egocentric strategies, we consider the question of optimal reorientation rates for the navigation of an agent moving along a preferred direction in the presence of multiple sources of noise. We address this using a model that takes the form of a correlated random walk at short time scales that is punctuated by reorientation events leading to a biased random walks at long time scales. This allows us to identify optimal alternation schemes and characterize their robustness in the context of noisy sensory acquisition as well as performance errors linked with variations in environmental conditions and agent-environment interactions.

Programming curvature using origami tessellations

Origami describes rules for creating folded structures from patterns on a flat sheet, but does not prescribe how patterns can be designed to fit target shapes. Here, starting from the simplest periodic origami pattern that yields one-degree-of-freedom collapsible structures-we show that scale-independent elementary geometric constructions and constrained optimization algorithms can be used to determine spatially modulated patterns that yield approximations to given surfaces of constant or varying curvature. Paper models confirm the feasibility of our calculations. We also assess the difficulty of realizing these geometric structures by quantifying the energetic barrier that separates the metastable flat and folded states. Moreover, we characterize the trade-off between the accuracy to which the pattern conforms to the target surface, and the effort associated with creating finer folds. Our approach enables the tailoring of origami patterns to drape complex surfaces independent of absolute scale, as well as the quantification of the energetic and material cost of doing so.

Biomimetic 4D printing

Shape-morphing systems can be found in many areas, including smart textiles, autonomous robotics, biomedical devices, drug delivery and tissue engineering. The natural analogues of such systems are exemplified by nastic plant motions, where a variety of organs such as tendrils, bracts, leaves and flowers respond to environmental stimuli (such as humidity, light or touch) by varying internal turgor, which leads to dynamic conformations governed by the tissue composition and microstructural anisotropy of cell walls. Inspired by these botanical systems, we printed composite hydrogel architectures that are encoded with localized, anisotropic swelling behaviour controlled by the alignment of cellulose fibrils along prescribed four-dimensional printing pathways. When combined with a minimal theoretical framework that allows us to solve the inverse problem of designing the alignment patterns for prescribed target shapes, we can programmably fabricate plant-inspired architectures that change shape on immersion in water, yielding complex three-dimensional morphologies.

Recovery of locomotion after injury in Drosophila melanogaster depends on proprioception

Locomotion is necessary for survival in most animal species. However, injuries to the appendages mediating locomotion are common. We assess the recovery of walking in Drosophila melanogaster following leg amputation. Whereas flies pre-amputation explore open arenas in a symmetric fashion on average, foreleg amputation induces a strong turning bias away from the side of the amputation. However, we find that unbiased walking behavior returns over time in wild-type flies, while recovery is significantly impaired in proprioceptive mutants. To identify the biomechanical basis of this locomotor impairment and recovery, we then examine individual leg motion (gait) at a fine scale. A minimal mathematical model that links neurodynamics to body mechanics during walking shows that redistributing leg forces between the right and left side enables the observed recovery. Altogether, our study suggests that proprioceptive input from the intact limbs plays a crucial role in the behavioral plasticity associated with locomotor recovery after injury.

The organization and control of an evolving interdependent population

Starting with Darwin, biologists have asked how populations evolve from a low fitness state that is evolutionarily stable to a high fitness state that is not. Specifically of interest is the emergence of cooperation and multicellularity where the fitness of individuals often appears in conflict with that of the population. Theories of social evolution and evolutionary game theory have produced a number of fruitful results employing two-state two-body frameworks. In this study, we depart from this tradition and instead consider a multi-player, multi-state evolutionary game, in which the fitness of an agent is determined by its relationship to an arbitrary number of other agents. We show that populations organize themselves in one of four distinct phases of interdependence depending on one parameter, selection strength. Some of these phases involve the formation of specialized large-scale structures. We then describe how the evolution of independence can be manipulated through various external perturbations.

A geometric model for the periodic undulation of a confined adhesive crack

Inspired by experiments on the instability of confined interfacial cracks, we construct a minimal mathematical model based on symmetry arguments that can reproduce the form of the crack front in a confined domain. We show that the model can be interpreted in terms of the buckling and post-buckling response of a compressed elastica with a nonuniform bending stiffness that is adhered to a linearly elastic substrate. The model has three parameters that allow us to capture the primary wavelength associated with the onset of an undulatory instability of a straight crack front, as well as the finger amplitudes and finger widths in the nonlinear development of the instability. We determine these parameters using an optimization procedure that minimizes the square error between the computed profile and experimental observations. The results of this procedure yield numerical solutions that agree well with the finger profiles experimentally observed in films of different thicknesses. Our approach shows the efficacy of simple models based on symmetry in explaining interfacial instabilities governed by different physical mechanisms.

Pressure-driven occlusive flow of a confined red blood cell

When red blood cells (RBCs) move through narrow capillaries in the microcirculation, they deform as they flow. In pathophysiological processes such as sickle cell disease and malaria, RBC motion and flow are severely restricted. To understand this threshold of occlusion, we use a combination of experiment and theory to study the motion of a single swollen RBC through a narrow glass capillary of varying inner diameter. By tracking the movement of the squeezed cell as it is driven by a controlled pressure drop, we measure the RBC velocity as a function of the pressure gradient as well as the local capillary diameter, and find that the effective blood viscosity in this regime increases with both decreasing RBC velocity and tube radius by following a power-law that depends upon the length of the confined cell. Our observations are consistent with a simple elasto-hydrodynamic model and highlight the role of lateral confinement in the occluded pressure-driven slow flow of soft confined objects.

Elastohydrodynamics and Kinetics of Protein Patterning in the Immunological Synapse

We propose a minimal mathematical model for the physical basis of membrane protein patterning in the immunological synapse (IS), which encompass membrane mechanics, protein binding kinetics and motion, and fluid flow in the synaptic cleft. Our theory leads to simple predictions for the spatial and temporal scales of protein cluster formation, growth and arrest as a function of membrane stiffness, rigidity and kinetics of the adhesive proteins, and the fluid flow in the synaptic cleft. Numerical simulations complement these scaling laws by quantifying the nucleation, growth and stabilization of proteins domains on the size of the cell. Direct comparison with experiment shows that passive elastohydrodynamics and kinetics of protein binding in the synaptic cleft can describe the short-time formation and organization of protein clusters, without evoking any active processes in the cytoskeleton. Despite the apparent complexity of the process, our analysis shows that just two dimensionless parameters characterize the spatial and temporal evolution of the protein pattern: a ratio of membrane elasticity to protein stiffness, and the ratio of a hydrodynamic time scale for fluid flow relative to the protein binding rate. A simple phase diagram encompasses the variety of patterns that can arise.

A proprioceptive neuromechanical theory of crawling

The locomotion of many soft-bodied animals is driven by the propagation of rhythmic waves of contraction and extension along the body. These waves are classically attributed to globally synchronized periodic patterns in the nervous system embodied in a central pattern generator (CPG). However, in many primitive organisms such as earthworms and insect larvae, the evidence for a CPG is weak, or even non-existent. We propose a neuromechanical model for rhythmically coordinated crawling that obviates the need for a CPG, by locally coupling the local neuro-muscular dynamics in the body to the mechanics of the body as it interacts frictionally with the substrate. We analyse our model using a combination of analytical and numerical methods to determine the parameter regimes where coordinated crawling is possible and compare our results with experimental data. Our theory naturally suggests mechanisms for how these movements might arise in developing organisms and how they are maintained in adults, and also suggests a robust design principle for engineered motility in soft systems.

The dynamics of sperm cooperation in a competitive environment

Sperm cooperation has evolved in a variety of taxa and is often considered a response to sperm competition, yet the benefit of this form of collective movement remains unclear. Here, we use fine-scale imaging and a minimal mathematical model to study sperm aggregation in the rodent genus Peromyscus. We demonstrate that as the number of sperm cells in an aggregate increase, the group moves with more persistent linearity but without increasing speed. This benefit, however, is offset in larger aggregates as the geometry of the group forces sperm to swim against one another. The result is a non-monotonic relationship between aggregate size and average velocity with both a theoretically predicted and empirically observed optimum of six to seven sperm per aggregate. To understand the role of sexual selection in driving these sperm group dynamics, we compared two sister-species with divergent mating systems. We find that sperm of Peromyscus maniculatus (highly promiscuous), which have evolved under intense competition, form optimal-sized aggregates more often than sperm of Peromyscus polionotus (strictly monogamous), which lack competition. Our combined mathematical and experimental study of coordinated sperm movement reveals the importance of geometry, motion and group size on sperm velocity and suggests how these physical variables interact with evolutionary selective pressures to regulate cooperation in competitive environments.